Force-Induced Unzipping Transitions in an Athermal CrowdedEnvironmentDavid L. Pincus and D. Thirumalai*

Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, United States

ABSTRACT: Using theoretical arguments and extensiveMonte Carlo (MC) simulations of a coarse-grained three-dimensional off-lattice model of a β-hairpin, we demonstratethat the equilibrium critical force, Fc, needed to unfold thebiopolymer increases nonlinearly with increasing volumefraction occupied by the spherical macromolecular crowdingagent. Both scaling arguments and MC simulations show thatthe critical force increases as Fc ≈ φc

α. The exponent α islinked to the Flory exponent relating the size of the unfoldedstate of the biopolymer and the number of amino acids. Thepredicted power law dependence is confirmed in simulations ofthe dependence of the isothermal extensibility and the fraction of native contacts on φc. We also show using MC simulations thatFc is linearly dependent on the average osmotic pressure (P) exerted by the crowding agents on the β-hairpin. The highlysignificant linear correlation coefficient of 0.99657 between Fc and P makes it straightforward to predict the dependence of thecritical force on the density of crowders. Our predictions are amenable to experimental verification using laser optical tweezers.

■ INTRODUCTIONThe study of the protein folding problem was galvanized byusing concepts from the physics of disordered systems. Using acoarse-grained description of folding, expressed in terms of anuncorrelated distribution of energies of protein conformationscorresponding to the values at local minima in a multidimen-sional energy landscape, Bryngelson and Wolynes1,2 mappedthe problem of equilibrium statistical mechanics of proteinfolding to a random energy model in which the native stateplays a special role. These influential works and subsequentstudies3 showed that most naturally evolved sequences arefoldable, which means that they reach the stable native state onbiologically relevant time scales. In this picture, foldablesequences are characterized by large differences in theenvironmental-dependent folding temperature (Tf) and theglass transition temperature (Tg) at which the kinetics becomesso sluggish that the folded state is inaccessible on biologicallyrelevant time scales. Related ideas rooted in polymer physicsfurther showed that the interplay of Tf, and the equilibriumcollapse temperature (Tθ)

4 could be used to not only fullycharacterize the phase diagram of generic protein sequences butalso determine their foldability, a prediction that has beenexperimentally validated very recently.5 In the interveningyears, an impressively large number of important theoreticaland experimental works (for a recent collection, see ref 6 andreferences cited therein) on a variety of seemingly unrelatedproblems associated with protein folding have appeared, thusgreatly expanding the scope and utility of concepts fromstatistical mechanics and polymer physics. Through thesedevelopments, an expansive view of protein folding and its rolein biophysics has emerged7 with current applications rangingfrom assisted folding8−11 to describing the functions of

molecular motors12−16 using models originally devised tounderstand protein folding kinetics.A particularly important problem that has benefited from the

focus on protein folding is the role molecular crowding plays inmodulating the thermodynamics and kinetics of folding ofproteins17 and RNA18−20 although its importance wasrecognized long ago.21 It is now widely appreciated that thecytosol is a crowded heterogeneous medium containing avariety of macromolecules such as ribosomes, lipids, and RNA.As a result, folding, diffusion, and other biological processes insuch an environment could be different from what transpiresunder infinite dilution conditions. The effects of macro-molecular crowding on the stability of synthetic as well asbiopolymers have been extensively investigated17,19−33 becauseof the potential relevance for folding under cellular conditions.In general, several interaction energy and length scalesdetermine whether crowding agents stabilize, have negligibleeffect, or even destabilize the folded states of proteins.29 As aresult a number of scenarios can emerge depending on thenature of crowding agents, and the choice of proteins. Thesimplest scenario arises when both the crowder−crowder andcrowder−protein interactions are dominated by excludedvolume. Although this situation may not accurately characterizeeven in vitro experiments, it has the advantage that folding inthis situation can be described using a combination of scalingarguments and simulations.29 Nonspecific athermal crowders

Special Issue: Peter G. Wolynes Festschrift

Received: March 25, 2013Revised: June 18, 2013Published: June 21, 2013

Article

pubs.acs.org/JPCB

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(only the excluded volume interactions between the crowdersand the crowder and the protein are relevant) tend to shift thefolded ⇌ unfolded (or equivalently the zipped ⇌ unzipped)equilibrium of a biopolymer toward the folded state by theentropic stabilization theory (EST), because this maximizes thefree-volume available (and hence entropy) to the crowdingagents. This simple theory is based on the elegant concept ofdepletion interaction,34−37 which posits that the crowdingparticles decrease the entropy of the unfolded state to a greaterextent than the folded state, thus differentially stabilizing theordered structure.29

The EST can be validated by measuring the dependence ofthe melting temperature on the volume fraction (φc) of thecrowding particles. If EST is valid then ΔTm(φc) = Tm(φc) −Tm(0) should increase. Indeed, absence of any change inΔTm(φc) indicates that enthalpic effects play an important role.Another way to quantify the extent of stabilization is to askwhat critical force (Fc) would be necessary to unfold abiopolymer at a given volume fraction (φc) of the crowdingagents. In this paper, we study the simple case of unzipping apolypeptide chain that forms a β-hairpin by applyingmechanical force as a function of volume fractions ofmonodisperse spherical crowding particles. The study of thezipping/unzipping of biopolymers has a rich history38−49 andhas even formed the basis of assessing folding mechanisms ofproteins.50

Surprisingly, there have been very few experimental51,52 ortheoretical studies18 investigating the effect of mechanical forceon proteins in a crowded environment. The experimentalstudies have argued that Fc increases linearly with φc, whereasthe theoretical arguments18 predict a nonlinear dependence,which was shown to provide a good fit to the experimental data.In this paper, we argue that the unzipping of a biopolymerunder constant tension could be consistent with lineardependence only for small φc. At higher volume fractions, Fcdoes increase nonlinearly with φc. The increase in Fc, relative toits value at φc = 0, linked to crowding-induced stability, arisesbecause of a depletion of the crowding particles from theproximity of proteins. This, in turn, results in the crowdingagents exerting an osmotic pressure on the biopolymer.Unzipping the biopolymer requires that the imposed tensionperform work against this osmotic pressure. Thus, it is naturalto assume that Fc should be linearly dependent on the averagepressure (P) associated with crowding particles modeled ashard spheres. We have verified this relation using extensiveMonte Carlo simulations, and we present a simple method fordetermining Fc at an arbitrary φc once the linear dependence ofFc on P is known.

■ METHODSModel. In order to explore the effects of crowding on the

unzipping of a biopolymer, we chose the 16 residue sequencethat forms a β-hairpin structure and had been previously usedto illustrate the effects of confinement on protein folding.48 Thestructure corresponds to the C-terminal β-hairpin of protein G(PDB accession ID 1GB1), a model system that has beenextensively studied using computations53−59 following an initialpioneering experimental study.50

In our simulations, we used a coarse-grained representationof the polypeptide chain. We modeled the hairpin as acollection of Np = 16 spheres of diameter σp = 0.38 nm (eachrepresenting a residue) with configuration {ri}i=1

Np , and crowdersas a monodisperse collection of Nc spheres of diameter σc = 1.0

nm with configuration {Rl}l=1Nc . The Hamiltonian depended on

both the positions of the crowders and the conformations ofthe polypeptide chain:

≡ + +

+ +

r R R r r R

r r

({ }, { }) ({ }) ({ }) ({ }, { })

({ }) ({ })

i l l i i l

i i

cc pp pc

bond coop (1)

The first three terms on the right-hand side (rhs) of eq 1accounted for nonbonded crowder−crowder (cc), protein−protein (pp), and protein−crowder (pc) interactions, respec-tively. The penultimate term on the rhs of eq 1( bond({ri}))was used to enforce chain connectivity, while the final term onthe rhs of eq 1 ( coop({ri})) was used to ensure that thehairpin underwent a cooperative unzipping transition undertension. The interactions between the crowding particles weretaken to be

∑ υ= | − |>

R R R({ }) ( )lJ l

l Jcc cc(2)

where

υσ

σ=

∞ ≤

>

⎪

⎪

⎧⎨⎩

rr

r( )

( )

0 ( )cc

c

c (3)

Similarly, we used hard-sphere potentials to model theinteractions between the crowders and the polypeptide (pc):

∑ υ= | − |r R R r({ }, { }) ( )i li l

l ipc,

pc(4)

where

υσ σ

σ σ=

∞ ≤ +

> +⎪⎪⎧⎨⎩

rr

r( )

( ( )/2)

0 ( ( )/2)pc

p c

p c (5)

The term pp({ri},{ri0}) in eq 1 was decomposed into native

(N) and non-native (NN) contributions by partitioning the setof residue−residue distances into those that were less than acutoff (Rcut = 0.8 nm) in the crystal structure and those greaterthan Rcut (i.e., {{|ri − rj|} ≡ {rij}={rij:|ri

0 − rj0| ≤ Rcut}∪{rij:|ri0 −

rj0| > Rcut}). Letting η = {rij:|ri

0 − rj0| ≤ Rcut} and ϑ = {rij:|ri

0 − rj0|

> Rcut}, we write

η= + ϑr r({ }, { }) ( ) ( )i ipp0

ppN

ppNN

(6)

∑η υ=η∈

d( ) ( )d

ppN

ppN

(7)

with

υ =

∞ <

−ϵ ≤ <

>

⎧⎨⎪⎪

⎩⎪⎪

d

d d

d d

d d

( )

( / 0.8)

(0.8 / 1.2)

0 ( / 1.2)

ppN

0

0

0(8)

where d0 is the value of d in the crystal structure.Similarly,

∑ υϑ =∈ϑ

d( ) ( )d

ppNN

ppNN

(9)

where

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υσ

σ=

∞ ≤

>⎪⎪⎧⎨⎩

dd

d( )

( )

0 ( )ppNN p

p (10)

Chain connectivity was enforced with a sum of box-like terms:

∑ υ= | − |<

+r r r({ }) ( )ii N

i ibond bond 1

p (11)

where

υ =

∞ <

≤ ≤

∞ >

⎧⎨⎪⎪

⎩⎪⎪

r

r r

r r

r r

( )

/ 0.8

0 0.8 / 1.2

/ 1.2

bond

b0

b0

b0

(12)

and rb0 is an ideal Cα−Cα “bonding” distance of 0.38 nm.

The cooperativity term ( coop({ri})) is a coarse-grainedrepresentation of hydrogen-bonding type interactions and has anearest-neighbor Ising-like character,

∑= − Θ − Θ −=

− −J d d d dr({ }) ( ) ( )il

l l l lcoop2

ncoop0

10

1(13)

where J = ϵ/5, Θ(x) is a Heaviside function, and dl (dl0) is the

distance (PDB distance) separating a pair of complementaryresidues in the strand (l and l − 1 denote nearest neighborpairs). There were ncoop = 7 pairs of complementary residuesin the strand with PDB numbering: {{41,56},{42,55},{43,54},{44,53},{45,52},{46,51},{47,50}} (see Figure 1 for the

numbering of the residues as well as the seqence). Note thatnot all of these residue pairs are hydrogen bonded in the nativehairpin. In general, strand pairs exist as parts of larger β-sheetsand make some hydrogen bonds between the strands of thepair as well as some hydrogen bonds with other strands of thesheet. The coarse-grained nature of eq 13 renders the modelsufficiently general to ensure transferability to models of RNAor DNA hairpins. Under such circumstances, eq 13 wouldmimic the stacking interactions, which are known to stabilizenucleic acids.

Simulation Methods. We used a standard Metropolisalgorithm to simulate the model described by eq 1 and toobtain thermodynamic quantities of interest. Crowder trialmoves were attempted in a “single-spin flip” manner andconsisted of random repositioning of a crowder through thegeneration of three independent and uniformly distributedrandom variables (rv’s) on the interval [−L/2,L/2], where L =29.7 nm is the length of a side of the cubic simulation box.The position of residue 1 of the hairpin was held fixed at the

origin throughout all simulations (i.e., r1(t) = 0 ∀t). Theremaining Np − 1 residue trial moves were randomly selectedfrom a set of two possibilities. One type of move correspondedto that used by Baumgar̈tner and Binder60 for simulating afreely jointed chain; a random angle γ was chosen from auniform distribution on [0,2π) and an attempt was made todisplace residue i by γ radians along the circle perpendicular tothe line connecting residues i − 1 and i + 1. For the residue atthe free-end of the chain two random angles (β,γ) were chosen,and an attempt was made to move the residue to a new pointon the sphere centered at residue Np − 1. The second type ofmove corresponded to a random change in the bond lengthconnecting residue i to residue i − 1 (i = 2, 3, ..., Np); a uniformrv, , on (0.8,1.2) was generated, and an attempt was made tomap ri |→ (ri − ri−1) + ri−1. A trial move from μ → ν wasaccepted with probability (A(μ → ν)):

μ ν→ =

−− − >ν μ

ν μ

− − −ν μ ν μ⎧⎨⎪

⎩⎪

A

F z z

( )

e e ( )( ) 0

1 otherwise

k T k T F z z( )/( ) (1/ ) ( )B B

(14)

where T is the temperature, kB is Boltzmann’s constant, ν and

μ are as above, F is the constant tension applied to thepolymer, and zν and zμ are the extension in state ν and μ,respectively, of the polymer in the direction of the appliedforce.

Data Collection. Time, measured in Monte Carlo steps(MCS), corresponded to the attempted displacement of (Nc +Np − 1) particles, since one end of the chain was always heldfixed to the origin. Data from a trajectory were collected every1000 MCS. Figure 2 reveals that this is significantly longer than

Figure 1. Snapshots of the simulated system in the absence ofmechanical force: (a) φc = 0.1 (Nc = 5000); (b) φc = 0.2 (Nc =10000); (c) φc = 0.3 (Nc = 15000); (d) φc = 0.4 (Nc = 20000). Thehairpin corresponds to the small dark spot at the center of each of theboxes. The center of the figure shows a blowup of the region adjacentto the hairpin. The purpose of showing the four snapshots is toillustrate that the biopolymer is jammed in a sea of crowding particles.The blowup in the center shows structure of the β-hairpin along withthe sequence numbering of the 16 residues.

Figure 2. Root mean square deviation (rmsd) of the crowding agentsfrom an equilibrated initial state as a function of time (τ, measured inMonte Carlo steps per free particle (MCS)) for trajectories at φc = 0.1(blue), φc = 0.2 (green), φc = 0.3 (orange), and φc = 0.4 (red). Notethat crowder trial moves are reasonably successful for φc ≤ 0.3, whichimplies that the allowed conformations are ergodically sampled. Theacceptance ratio for such trial moves is significantly reduced at φc = 0.4although the errors in the results are small as indicated by consistencybetween different measures. The sampling interval used for collectingthe data presented in all figures below was 1000 MCS.

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the time required for the RMSD of the crowding agents froman equilibrated initial state to plateau at all volume fractionsexcept φc = 0.4. Even at φc = 0.4, the RMSD has increasedsubstantially after 1000 MCS. We used 0, 5000, 10000, 15000,and 20000 crowders to simulate crowder volume fractions of0.0, 0.1, 0.2, 0.3, and 0.4, respectively. For each φc, data wascollected at tensions between 0 pN and 40 pN at 1 pNintervals. Snapshots of simulations at each of the nonzero φcand in the absence of tension are illustrated in Figure 1. Data ateach force and each φc was collected from multiple trajectoriesstarting from previously equilibrated configurations (in turnbased on trajectories initiated from random initial crowderconfigurations at both high- and low-force hairpin config-urations).

■ RESULTSRadial Distribution between Crowders and the

Hairpin. Figure 3 is a plot of the radial distribution (g(r)) of

crowders about the center of mass of the hairpin versusdistance (r) (i.e., g(r) = V/[Nc]⟨∑lδ(r − (Rl − rcm))⟩). Themaxima in these plots correspond to the average diameter ofthe region to which the hairpin finds itself confined (D).Interestingly, the plots illustrate that the average size of theregion is inversely proportional to the crowder density (i.e., D≈ φc

−1). This suggests that, perhaps, the region in which thehairpin on average is localized is aspherical.61 If the region werespherical, we would expect that D ≈ φc

−1/3.

The observation that D ≈ φc−1 in conjunction with an

approximate mapping between crowding and confinementcould be used to obtain the expected scaling of the dependenceof the critical force required to the unfold the β-hairpin, Fc, onφc. Because the confining region is described by a single length,D, the EST can be used to identify the enhancement in thestability of the ordered state with the loss in entropy of theunfolded state upon confinement. A similar scaling approach,using concepts developed in the context of polymer physics, hasbeen used to study confinement effects on biopolymers.48,62−64

Using this inherently mean-field argument, we expect

φ≈ Δ Δ ≈Δ

≈ν 퇇F T S x R D

k Tx

A/ ( / )uu

c g1/ B

c1/

(15)

In the above equation, TΔS is the penalty for confining thepolypeptide chain with dimension Rg in a region with size D,Δxu‡ is the minimum extension needed to unfold the protein,and ν is the Flory exponent. Because D ≈ φc

−1, we expect thatFc ≈ φc

1/ν ≈ φc5/3 assuming that ν ≈ 0.6. If D ≈ φc

−1/3, aswould be the case if the unfolded state were spherical, then itfollows that Fc ≈ φc

5/9, a result that we derived previously18 toanalyze the experimental data on forced unfolding of ubiquitin.

Numerical Evidence for Equation 15. Plots of theaverage extension of the hairpin (⟨z⟩) versus applied tension(F) presented in Figure 4a show that the ⟨z⟩ decreasesmonotonically with φc at moderate values of F. This impliesthat crowding in essence decreases ⟨z⟩ because the entropicpenalty to stretch a protein in a crowded environment is far toolarge. In other words, the probability of finding a region free ofcrowders decreases exponentially as the extension increases,which explains the observed results in Figure 4a. Theisothermal extensibility (χ ≡ ∂⟨z⟩/∂F) plots in Figure 4breveal that Fc (i.e., the value of F at which χ is a maximum)increases monotonically with increasing φc. A plot of Fc versusφc (Figure 5a) subsequently revealed that the power-lawdependence of Fc on φc is characterized by an exponent α ≅1.6, which is in accord with the scaling predictions in eq 15.Data collapse of χ based on a scaling function X((F − Fc)/Fc)that is independent of Nc revealed that χ ≈ (1 − ANc

dχ), wheredχ ≅ 1.43 and A ≅ 1.7 × 10−7. This shows that the effects ofcrowding and force can be separated, which to some extentjustifies the scaling theory predictions. Thus, when measured interms of the reduced distance to the critical force, the primaryeffect of the crowders is to decrease the extensibility of thechain.We can also obtain the dependence of Fc on φc using the F-

dependent changes in an order parameter that characterizes the

Figure 3. Radial distribution function (g(r)) of the crowders about thecenter of mass of the hairpin at various volume fractions and at F = 0pN. Red squares, orange diamonds, green upward triangles, and bluedownward triangles, respectively, correspond to φc = 0.1, 0.2, 0.3, and0.4. The maxima in g(r) lie at r = 15.5, 14.5, 13.5, and 12.5 Å,respectively. This suggests that the size of the region to which thehairpin is confined (D) is inversely proportional to φc, implying thatthis region is aspherical. The symbols represent raw data; curvescorrespond to smoothing using eq 6.48 of Allen and Tildesley.66

Figure 4. Plot of (a) average extension (⟨z⟩) as a function of force (F) and (b) isothermal extensibility (χ ≡ ∂⟨z⟩/∂F|T) versus force (F). Black, red,orange, green, and blue curves, respectively, correspond to volume fractions (φc) of 0, 0.1, 0.2, 0.3, and 0.4. All curves were calculated using themultiple histogram reweighting method; symbols in plot a correspond to unreweighted data.

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folded state. The extent of structure formation can be inferredusing the average fraction of native contacts, ⟨Q⟩. In Figure 6,

we show ⟨Q⟩ as a function of F. For all values of F, thecrowding particles increase ⟨Q⟩, which is a reflection of theenhanced stabilization of the native state of β-hairpin at φc ≠ 0.Let us define Fm using ⟨Q⟩ = 0.5 at φc = 0. At this value of Fm,Figure 6a shows that ⟨Q⟩ ≈ 0.75 at φc = 0.4. The critical force,Fc, can identified with the force at which |d⟨Q⟩/dF| (Figure 6b)achieves a maximum. It is clear that Fc is an increasing functionof φc (Figure 6c). Just as in Figure 4a, where Fc is identified

with the maximum in the isothermal extensibility, we find thatFc ≈ φc

α with α ≈ 1.6 (Figure 6c). The numerical simulationsusing different measures confirm the scaling predictionsshowing the power law increase in the φc-dependent criticalforce required to rupture the hairpin.

Osmotic (or Disjoining) Pressure Explains the Originof φc-Dependent Fc. Insights into our results can be obtainedby viewing the depletion forces from a different perspective.Because of the repulsive interaction between the crowders andthe polypeptide chain, the crowding particles are depleted fromthe surface of the protein. In the process, the crowding particlesnot only gain translational entropy, but they also exert anosmotic pressure on the polypeptide chain, thus forcing it toadopt a compact structure. In other words, the crowders can beviewed as providing an isothermal and isobaric bath for thehairpin. In such a case, it is natural to assume that Fc isproportional to the average pressure (P) associated with a hardsphere fluid at that density and volume fraction:

= +F mP bc (16)

where m and b are constants to be determined.The disjoining or osmotic pressure can, in turn, be calculated

from the contact value g(σ) ≡ limr→σ+g(r) of the crowder−

crowder radial distribution function using the standard relation,

ρ φ σ= +P k T g(1 4 ( ))B c (17)

Equation 17 follows from the virial-based expression for hardsphere systems,

∫ρπρ υ= −

∞Pk T k T

g rr

r r12

3( )

dd

dB B 0

3

(18)

via the substitution g(r) = ψ(r) e−βυ(r) and by noting that theBoltzmann factor e−βυ = θ(r − σ) for hard spheres where θ(x) isthe step function.From the Figure 7a showing the crowder−crowder g(r) at

volume fractions φc = 0.1, 0.2, 0.3, and 0.4, we computed g(σ),which was subsequently used to determine the average pressureat each φc. The linear correlation coefficient (r) between thetwo variables (F and P) was determined to be 0.99657 (Figure7b). The probability that five measurements of twouncorrelated random variables would yield a correlationcoefficient this high is 2Γ(2)/[√πΓ(3/2)]∫ 0.99657

1 (1 − x2)1/2

dx = 0.00024, where Γ(x) is Euler’s gamma function. In Figure7b, we provide the best fit line to the data yielding m = 0.24nm2 and b = 13.84 pN. Thus, Fc is linearly related to theosmotic pressure arising from depletion forces, whose strengthis a measure of the stabilization of the ordered state.

Figure 5. (a) Critical force (Fc) of the hairpin versus crowder volume fraction (φc). Fc displays a power law dependence on the volume fraction ofcrowding agent with an exponent (α) of 1.55. (b) Data collapse of the isothermal extensibility (χ) (Figure 4) shows that χ ≈ (1 − ANc

dχ)X((F − Fc)/Fc), where the scaling function (X(x)) is independent of the number of crowders (Nc). Thus, the dependence of the isothermal extensibility on Nc ischaracterized by an exponent (dχ) of 1.43. Black, red, orange, green, and blue curves, respectively, are for Nc = 0, 5000, 10000, 15000, and 20000.

Figure 6. (a) Average fraction of native contacts (⟨Q⟩) versus appliedtension (F). (b) Absolute value of d⟨Q⟩/dF versus F. The force thatmaximizes |d⟨Q⟩/dF| at a particular volume fraction φc corresponds tothe critical force Fc(φc) at φc. (c) A plot of Fc versus φc verifies theresults illustrated in Figure 5a; Fc ≈ φc

α where α ≅ 1.6.

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In order to obtain the dependence of Fc on φc, a reliablerelationship between P and φc needs to be established.Although P can be calculated using simulations, it would beconvenient to obtain approximate analytically calculableestimates of P. The average pressure associated with the hardspheres can be determined at all φc using the successfulsemiempirical Carnahan−Starling equation of state:

ρφ φ φ

φ=

+ + −−

Pk T

1

(1 )B

c c2

c3

c3

(19)

Figure 7c shows Fc versus φc (blue circles), as well as the curveassociated with the best-fit linear relation between Fc and Pcalculated using eq 19. This linear relation yielded m = 0.17nm2 and b = 13.96 pN, which are close to the values obtainedby fitting Fc to numerically computed values for P. The starsillustrate (φc,Fc) ordered pairs associated with the best-fit linearrelation between Fc and P as calculated from eq 17 (as in Figure7b). We surmise that P approximated by eq 19 can be used toobtain accurate estimates of Fc given knowledge of thecoefficients m and b.Finally, the accuracy of the Carnahan−Starling equation is

assessed by plotting in Figure 7d eq 19 as well as severaltruncated Taylor-series expansions:

∑πσ

φ≅=

Pk T

a6

i

n

iiB

31

c(20)

of this equation of state. Note that the relative error associatedwith the linear approximation (n = 1) of 0.342651 is large at φc= 0.1, while the relative error is only 0.0326804 at φc = 0.4when n = 7 terms are included in the expansion. The

coefficients of the expansions are a1 = 1, a2 = 4, a3 = 10, a4 = 18,a5 = 28, a6 = 40, and a7 = 54. The linear relation between Fcand P shows that close to φc = 0, Fc should depend only linearlyon φc because P is approximately linearly dependent on φc forsmall φc. However, in order to determine the value of Fc at anarbitrary φc, one should first determine the appropriate linearrelation between Fc and P. The critical force Fc can then bedetermined at an arbitrary φc using the linear relation andapproximate estimate of P given in eq 19.

■ CONCLUSIONSUsing simple theoretical arguments and extensive MCsimulations of a three-dimensional off-lattice model, we havedemonstrated for the first time that the critical force forunzipping a biopolymer under tension obeys a nonlineardependence on the volume fraction of crowding agent. Thisdependence can be characterized by a power law dependencewith an exponent α ≅ 1.6: Fc ≈ φc

α. The exponent α issurprisingly close to the scaling prediction 1/ν with ν ≈ 3/5.The numerical findings and scaling predictions can be

understood by noting that the crowders provide an isobaricenvironment for the protein. The osmotic pressure arises fromthe depletion forces due to expulsion of the crowding particlesfrom the protein and is entropic in origin. Because of theosmotic pressure, unzipping requires that the tension imposedon the hairpin perform mechanical work against the isotropicpressure. These arguments are fully confirmed in simulations,which demonstrate that Fc has a highly significant linearcorrelation with the pressure (P) of the hard-sphere crowdingparticles in which it is embedded. To determine Fc at anarbitrary φc, one should first determine the linear dependenceof Fc on P. The exact relation connecting Fc to φc then follows

Figure 7. (a) Crowder−crowder radial distribution fucntion (g(r)) versus separation distance (r/σ). Red squares, orange diamonds, green uptriangles, and blue down triangles, respectively, correspond to φc = 0.1, 0.2, 0.3, and 0.4. (b) The contact value g(σ) ≡ limr→σ

+g(r) from (a) was usedto calculate the average pressure (P) of the hard spheres at each φc using the virial derived equation P = ρkBT(1 + 4φcg(σ)). A plot of Fc versus Prevealed a highly significant correlation with a linear correlation coefficient r = 0.99657. Blue circles correspond to measured data, and the black solidline corresponds to the best fit line Fc = mP + b, where m = 0.24 nm2 and b = 13.84 pN. (c) The dependence of Fc on φc. Blue circles correspond tosimulation data. The solid black curve corresponds to the best-fit line relating Fc to the pressure (P(φc)) at φc, where P was calculated from thesemiempirical Carnahan−Starling equation of state: P = ρkBT(1 + φc + φc

2 − φc3)/(1 − φc)

3. The red stars also correspond to a best-fit line relatingFc to P(φc), where P was calculated as in panel b. (d) P versus φc as calculated via the Carnahan−Starling equation of state (black) and aftertruncating a Taylor-series expansion (P ≅ 6kBT/(πσ

3∑i = 1n aiφc

i) of this equation of state about φc = 0 after n = 1 (red), 2 (orange), 3 (yellow), 4(green), 5 (cyan), 6 (blue), and 7 (purple) terms.

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from the Carnahan−Starling equation of state. This relationshipshows that Fc displays an approximately linear dependence onφc for volume fractions near φc = 0. However, when examinedover a large range of φc, we expect that Fc should increasenonlinearly with φc as indicated by the scaling predictionsexploiting the relationship between crowding and confinement.Two comments about the scaling predictions are important

to make. (i) The exponent α relating the increase in Fc to φc,although related to the Flory exponent (ν), is likely to dependboth on the nature of the unfolded states of the protein and theshape of the crowding particles. If the overall shape of theunfolded state is nonspherical as is clearly the case for thehairpin (Figure 1) then α ≈ 1.6. On the other hand, if theunfolded state is spherical on average, as is likely to be the casefor larger proteins, then it is likely α ≈ 5/9, as arguedpreviously.18 (ii) The theoretical predictions are based on amean-field picture in which it is assumed that crowding(modeled with hard spheres) results in the protein beinglocalized to a cavity. Thus, fluctuations in the crowding particlesare ignored. These, especially close to the protein, could havesignificant effects. The good agreement between scalingpredictions and simulations suggests that the fluctuation effectsare not significant, at least for the case tested here. In principle,the importance of fluctuations can be tested by fixing thelocations of the crowding particles. Such quenched simulationsare equivalent to the present annealed simulations for theproperties of the proteins because in a large sample containingfixed obstacles the protein would sample many distinctenvironments. This is then the same as performing annealedsimulations. Therefore, we expect that the scaling propertiespredicted and tested here will not change even if thesimulations are done by fixing the locations of the crowdingparticles. Additional simulations on proteins, rather thanpolypeptide chains forming secondary structures, would beneeded to obtain accurate values of α.There are only very few experiments probing the limits of

mechanical stability of proteins in the presence of crowdingagents. For example, atomic force microscopy has been used toinvestigate the effects of dextran on the mechanical stability ofproteins.52 These researchers found Fc ≈ φc for φc ∈ [0.0,0.3]and nonlinearity only for φc > 0.3. It is important to note thattheir experimental setup is of an inherently nonequilibriumcharacter; one end of a protein is extended at a constant speed,while the other end is used to probe the chain’s tension.Furthermore, the protein examined (ubiquitin) is unlikely to bea two-state folder and may undergo distinct unzipping reactionsat multiple tensions. In this case, the effects of crowding in anenergy landscape with multiple barriers65 may have to bestudied. Because of the nonequilibrium nature of the AFMsetup, it would be desirable to verify the predictions of thepresent work using laser optical tweezer experiments in whichsmall constant forces can be applied.

■ AUTHOR INFORMATIONCorresponding Author*E-mail address: thirum@umd.edu.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe are grateful to the National Institutes of Health (GrantGM089685) for supporting the research. The authors kindly

thank the National Energy Research Scientific Computing(NERSC) Center for significant computational time andresources.

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